Abstract Consider a (2, 1) random walk in an i.i.d. random environment, whose environment involves certain parameter. We construct an M-estimator for the environment parameter which can be written as functionals of a multitype branching process with immigration in a random environment (BPIRE). Because the offspring distributions of the involved multitype BPIRE are of the linear fractional type, the limit invariant distribution of the multitype BPIRE can be computed explicitly. As a result, we get the weak consistency of the M-estimator. Our result is a generalization of Comets et al. [Stochastic Process. Appl. 2014, 124, 268–288].

中文翻译：

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参数随机环境中(2, 1)随机游走的M-estimator及其弱一致性

摘要 考虑在 iid 随机环境中的 (2, 1) 随机游走，其环境涉及某个参数。我们为环境参数构建了一个 M 估计器，它可以写成在随机环境中迁移的多类型分支过程（BPIRE）的泛函。因为所涉及的多类型 BPIRE 的后代分布是线性分数类型，所以可以明确计算多类型 BPIRE 的极限不变分布。结果，我们得到了 M-estimator 的弱一致性。我们的结果是 Comet 等人的概括。[随机过程。应用程序 2014, 124, 268–288]。